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In the classic proof of Bertrand's postulate by Paul Erdős, he shows that $x\# < 4^x$ where $x\#$ is the primorial for $x$.

Is there any tighter upper bound for a given primorial $x\#$? Ideally, does anyone know if there are any relatively recent papers on estimating the upper bound of a primorial?

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2  
From a little back-of-envelope calculation and testing, it looks like $e^x$ might be a pretty sharp bound. I expect (well, maybe, maybe not) you can show $x\# < 3^x$ with only tedium. –  Daniel Fischer Jul 23 '13 at 19:01
6  
@DanielFischer Chebyshev showed by an elementary argument that $x\#\ll a^x$ for every $a>2^{14/25}3^{9/25}5^{1/5}\approx 3.020887$. This might be the best bound that can be shown without substantially more work. –  Julian Rosen Jul 23 '13 at 19:21
    
@PinkElephants The value for $a$ looks as though something counting multiples of $2,\,3,\, 5$ was used. If that's the case, I can imagine including $7$ would push the value below $3$ (at the price of much more tedium). Anyway, thanks for the notification. Nice to know. –  Daniel Fischer Jul 23 '13 at 19:25
2  
@DanielFischer You're right about the possibility of improvement. It turns out Diamond and Erdos proved in 1980 that for any $a>e$, Chebyshev's method can be refined to show $\#x\ll a^x$. –  Julian Rosen Jul 23 '13 at 19:43
    
@PinkElephants Nice, thanks. –  Daniel Fischer Jul 23 '13 at 19:50

2 Answers 2

up vote 8 down vote accepted

A commonly studied arithmetic function is $$ \vartheta(x):=\sum_{p\leq x}\log(p)=\log(x\#). $$

The Prime Number Theorem can equally well be stated in terms of $\pi(x)$, $\psi(x)$, and $\vartheta(x)$. The equivalence of the statements is not difficult to show.

The classical error term for the Prime Number Theorem implies that there are constants $C_1,C_2>0$ such that $$ x\#<e^x e^{C_1x\exp\left(-C_2\sqrt{\log(x)}\right)}. $$ Basically any improvement on the error term in PNT improves this upper bound.

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For a fairly pleasant version, for $x>1,$ we have $$ \log(x \#) < x \left(1 + \frac{1}{2 \log x} \right). $$ This is Theorem 4, formula 3.15 in Rosser and Schoenfeld (1962), Illinois J. Math.

Then, for $x \geq 563,$ we have $$ \log(x \#) > x \left(1 - \frac{1}{2 \log x} \right). $$ This is Theorem 4, formula 3.14.

The title of the article is Approximate Formulas for some Functions of Prime Numbers. It is pages 64-94, volume 6. I think I just found it online at some point...

YES, LINK

I did a simple experiment with C++, each line is just the low R-S bound, then a prime p, then log(p#), then the high R-S bound.

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

   prime fac     R-S low           p     log(p#)     R-S high
           1    0.557305           2    0.693147      3.4427
           2     1.63464           3     1.79176     4.36536
           3     3.44666           5      3.4012     6.55334
           4     5.20136           7     5.34711     8.79864
           5     8.70632          11       7.745     13.2937
           6     10.4658          13       10.31     15.5342
           7     13.9999          17     13.1432     20.0001
           8     15.7736          19     16.0876     22.2264
           9     19.3323          23     19.2231     26.6677
          10     24.6939          29     22.5904     33.3061
          11     26.4863          31     26.0244     35.5137
          12     31.8766          37     29.6353     42.1234
          13     35.4797          41     33.3489     46.5203
          14     37.2837          43     37.1101     48.7163
          15     40.8963          47     40.9602     53.1037
          16     46.3254          53     44.9305     59.6746
          17     51.7652          59      49.008     66.2348
          18     53.5807          61     53.1189     68.4193
          19     59.0327          67     57.3236     74.9673
          20     62.6719          71     61.5863     79.3281
          21     64.4928          73     65.8768     81.5072
          22       69.96          79     70.2462       88.04
          23     73.6084          83      74.665     92.3916
          24     79.0861          89     79.1537     98.9139
          25     86.3982          97     83.7284     107.602
          26     90.0577         101     88.3435     111.942
          27     91.8882         103     92.9782     114.112
          28     95.5508         107     97.6511     118.449
          29     97.3829         109     102.342     120.617
          30     101.048         113      107.07     124.952
          31     113.892         127     111.914     140.108
          32     117.565         131     116.789     144.435
          33     123.077         137     121.709     150.923
          34     124.915         139     126.644     153.085
          35     134.112         149     131.648     163.888
          36     135.952         151     136.665     166.048
          37     141.475         157     141.721     172.525
          38         147         163     146.815         179
          39     150.685         167     151.933     183.315
          40     156.215         173     157.086     189.785
          41     161.747         179     162.274     196.253
          42     163.591         181     167.472     198.409
          43     172.817         191     172.724     209.183
          44     174.663         193     177.987     211.337
          45     178.356         197      183.27     215.644
          46     180.203         199     188.564     217.797
          47     191.287         211     193.915     230.713
          48     202.379         223     199.323     243.621
          49     206.078         227     204.747     247.922
          50     207.928         229     210.181     250.072
          51     211.628         233     215.632     254.372
          52     217.179         239     221.109     260.821
          53      219.03         241     226.594      262.97
          54     228.287         251     232.119     273.713
          55     233.843         257     237.668     280.157
          56     239.401         263      243.24     286.599
          57     244.959         269     248.835     293.041
          58     246.813         271     254.437     295.187
          59     252.373         277     260.061     301.627
          60     256.081         281     265.699     305.919
          61     257.936         283     271.345     308.064
          62     267.209         293     277.025     318.791
          63     280.196         307     282.752     333.804
          64     283.908         311     288.492     338.092
          65     285.765         313     294.238     340.235
          66     289.477         317     299.997     344.523
          67     302.476         331     305.799     359.524
          68     308.049         337     311.619     365.951
          69     317.338         347     317.468     376.662
          70     319.197         349     323.323     378.803
          71     322.914         353      329.19     383.086
          72      328.49         359     335.073      389.51
          73     335.927         367     340.979     398.073
          74     341.505         373       346.9     404.495
          75     347.084         379     352.838     410.916
          76     350.804         383     358.786     415.196
          77     356.385         389     364.749     421.615
          78     363.828         397     370.733     430.172
          79      367.55         401     376.727      434.45
          80     374.994         409     382.741     443.006
          81     384.302         419     388.779     453.698
          82     386.164         421     394.821     455.836
          83     395.475         431     400.887     466.525
          84     397.337         433     406.958     468.663
          85     402.925         439     413.043     475.075
          86      406.65         443     419.136      479.35
          87     412.239         449     425.243     485.761
          88     419.692         457     431.368     494.308
          89     423.419         461     437.501     498.581
          90     425.282         463     443.639     500.718
          91      429.01         467     449.785      504.99
          92     440.194         479     455.957     517.806
          93     447.651         487     462.145     526.349
          94     451.381         491     468.342     530.619
          95      458.84         499     474.554      539.16
          96      462.57         503     480.775      543.43
          97     468.165         509     487.007     549.835
          98     479.358         521     493.263     562.642
          99     481.224         523     499.523     564.776
         100     498.019         541     505.816     583.981
         101     503.618         547     512.121     590.382
         102     512.951         557     518.443     601.049
         103     518.552         563     524.777     607.448
         104     524.154         569      531.12     613.846
         105     526.021         571     537.468     615.979
         106     531.623         577     543.826     622.377
         107     540.961         587     550.201     633.039
         108     546.564         593     556.586     639.436
         109     552.168         599     562.981     645.832
         110     554.037         601      569.38     647.963
         111     559.641         607     575.788     654.359
         112     565.246         613     582.207     660.754
         113     568.983         617     588.631     665.017
         114     570.852         619      595.06     667.148
         115     582.065         631     601.507     679.935
         116      591.41         641      607.97      690.59
         117     593.279         643     614.436     692.721
         118     597.018         647     620.908     696.982
         119     602.626         653      627.39     703.374
         120     608.235         659     633.881     709.765
         121     610.105         661     640.374     711.895
         122     621.324         673     646.886     724.676
         123     625.064         677     653.404     728.936
         124     630.675         683      659.93     735.325
         125     638.156         691     666.469     743.844
         126     647.509         701     673.021     754.491
         127     654.992         709     679.585     763.008
         128     664.347         719     686.163     773.653
         129     671.832         727     692.752     782.168
         130     677.446         733     699.349     788.554
         131      683.06         739     705.954      794.94
         132     686.803         743     712.565     799.197
         133      694.29         751     719.186      807.71
         134     699.906         757     725.816     814.094
         135     703.649         761      732.45     818.351
         136     711.138         769     739.095     826.862
         137     714.882         773     745.746     831.118
         138     727.989         787     752.414     846.011
         139     737.352         797     759.095     856.648
         140     748.589         809      765.79     869.411
         141     750.462         811     772.489     871.538
         142     759.827         821     779.199     882.173
         143     761.701         823     785.912     884.299
         144     765.447         827      792.63     888.553
         145      767.32         829      799.35      890.68
         146     776.688         839     806.082     901.312
         147     789.803         853     812.831     916.197
         148     793.551         857     819.585     920.449
         149     795.425         859      826.34     922.575
         150     799.173         863     833.101     926.827
         151     812.291         877     839.877     941.709
         152      816.04         881     846.658      945.96
         153     817.914         883     853.442     948.086
         154     821.663         887      860.23     952.337
         155     840.408         907      867.04     973.592
         156     844.158         911     873.854     977.842
         157     851.657         919     880.678     986.343
         158     861.032         929     887.512     996.968
         159     868.533         937     894.354     1005.47
         160     872.283         941     901.201     1009.72
         161     877.909         947     908.055     1016.09
         162     883.535         953     914.914     1022.46
         163     896.665         967     921.788     1037.34
         164     900.416         971     928.667     1041.58
         165     906.043         977     935.551     1047.96
         166     911.671         983     942.442     1054.33
         167     919.175         991     949.341     1062.82
         168     924.803         997     956.245      1069.2
         169     936.061        1009     963.162     1081.94
         170     939.813        1013     970.083     1086.19
         171     945.443        1019     977.009     1092.56
         172     947.319        1021     983.938     1094.68
         173     956.702        1031     990.876      1105.3
         174     958.579        1033     997.816     1107.42
         175     964.209        1039     1004.76     1113.79
         176     973.593        1049     1011.72     1124.41
         177      975.47        1051     1018.68     1126.53
         178     984.855        1061     1025.64     1137.15
         179     986.732        1063     1032.61     1139.27
         180     992.363        1069     1039.59     1145.64
         181     1009.26        1087     1046.58     1164.74
         182     1013.01        1091     1053.57     1168.99
         183     1014.89        1093     1060.57     1171.11
         184     1018.65        1097     1067.57     1175.35
         185     1024.28        1103     1074.57     1181.72
         186     1029.91        1109     1081.59     1188.09
         187     1037.42        1117      1088.6     1196.58
         188     1043.06        1123     1095.63     1202.94
         189     1048.69        1129     1102.66     1209.31
         190     1069.35        1151     1109.71     1232.65
         191     1071.23        1153     1116.76     1234.77
         192     1080.62        1163     1123.81     1245.38
         193     1088.13        1171     1130.88     1253.87
         194     1097.53        1181     1137.95     1264.47
         195     1103.16        1187     1145.03     1270.84
         196      1108.8        1193     1152.12      1277.2
         197     1116.31        1201     1159.21     1285.69
         198     1127.59        1213     1166.31     1298.41
         199     1131.35        1217     1173.41     1302.65
         200     1136.98        1223     1180.52     1309.02
         201     1142.62        1229     1187.64     1315.38
         202      1144.5        1231     1194.75      1317.5
         203     1150.14        1237     1201.87     1323.86
         204     1161.41        1249        1209     1336.59
         205     1170.81        1259     1216.14     1347.19
         206     1187.73        1277     1223.29     1366.27
         207     1189.61        1279     1230.45     1368.39
         208     1193.37        1283      1237.6     1372.63
         209     1199.01        1289     1244.77     1378.99
         210     1200.89        1291     1251.93     1381.11
         211     1206.53        1297      1259.1     1387.47
         212     1210.29        1301     1266.27     1391.71
         213     1212.17        1303     1273.44     1393.83
         214     1215.93        1307     1280.61     1398.07
         215     1227.21        1319      1287.8     1410.79
         216     1229.09        1321     1294.99     1412.91
         217     1234.73        1327     1302.18     1419.27
         218      1266.7        1361     1309.39      1455.3
         219     1272.34        1367     1316.61     1461.66
         220     1277.98        1373     1323.84     1468.02
         221      1285.5        1381     1331.07      1476.5
         222     1302.43        1399     1338.31     1495.57
         223     1311.84        1409     1345.56     1506.16
         224        1325        1423     1352.82        1521
         225     1328.77        1427     1360.09     1525.23
         226     1330.65        1429     1367.35     1527.35
         227     1334.41        1433     1374.62     1531.59
         228     1340.05        1439     1381.89     1537.95
         229     1347.58        1447     1389.17     1546.42
         230     1351.34        1451     1396.45     1550.66
         231     1353.23        1453     1403.73     1552.77
         232     1358.87        1459     1411.01     1559.13
         233     1370.16        1471     1418.31     1571.84
         234     1379.57        1481     1425.61     1582.43
         235     1381.45        1483     1432.91     1584.55
         236     1385.21        1487     1440.21     1588.79
         237      1387.1        1489     1447.52      1590.9
         238     1390.86        1493     1454.83     1595.14
         239     1396.51        1499     1462.14     1601.49
         240      1407.8        1511     1469.46      1614.2
         241     1419.09        1523     1476.79     1626.91
         242     1426.62        1531     1484.12     1635.38
         243     1437.91        1543     1491.47     1648.09
         244     1443.56        1549     1498.81     1654.44
         245     1447.32        1553     1506.16     1658.68
         246     1452.97        1559     1513.51     1665.03
         247      1460.5        1567     1520.87      1673.5
         248     1464.27        1571     1528.23     1677.73
         249      1471.8        1579     1535.59      1686.2
         250     1475.56        1583     1542.96     1690.44
         251     1488.74        1597     1550.33     1705.26
         252     1492.51        1601     1557.71     1709.49
         253     1498.16        1607      1565.1     1715.84
         254     1500.04        1609     1572.48     1717.96
         255      1503.8        1613     1579.86      1722.2
         256     1509.45        1619     1587.25     1728.55
         257     1511.34        1621     1594.64     1730.66
         258     1516.99        1627     1602.04     1737.01
         259      1526.4        1637     1609.44      1747.6
         260     1545.23        1657     1616.85     1768.77
         261     1550.88        1663     1624.27     1775.12
         262     1554.65        1667     1631.69     1779.35
         263     1556.53        1669     1639.11     1781.47
         264     1579.14        1693     1646.54     1806.86
         265      1582.9        1697     1653.98      1811.1
         266     1584.79        1699     1661.42     1813.21
         267      1594.2        1709     1668.86      1823.8
         268     1605.51        1721     1676.31     1836.49
         269     1607.39        1723     1683.76     1838.61
         270     1616.81        1733     1691.22     1849.19
         271     1624.35        1741     1698.68     1857.65
         272        1630        1747     1706.15        1864
         273     1635.65        1753     1713.62     1870.35
         274      1641.3        1759     1721.09      1876.7
         275     1658.26        1777     1728.57     1895.74
         276     1663.91        1783     1736.06     1902.09
         277     1667.68        1787     1743.55     1906.32
         278     1669.56        1789     1751.04     1908.44
         279     1680.87        1801     1758.53     1921.13
         280     1690.29        1811     1766.03     1931.71
         281      1701.6        1823     1773.54      1944.4
         282     1709.14        1831     1781.05     1952.86
         283     1724.22        1847     1788.58     1969.78
         284     1737.41        1861      1796.1     1984.59
         285     1743.06        1867     1803.64     1990.94
         286     1746.83        1871     1811.17     1995.17
         287     1748.72        1873     1818.71     1997.28
         288     1752.49        1877     1826.24     2001.51
         289     1754.37        1879     1833.78     2003.63
         290      1763.8        1889     1841.33      2014.2
         291     1775.11        1901     1848.88     2026.89
         292     1780.76        1907     1856.43     2033.24
         293     1786.42        1913     1863.99     2039.58
         294     1803.39        1931     1871.55     2058.61
         295     1805.27        1933     1879.12     2060.73
         296     1820.35        1949     1886.69     2077.65
         297     1822.24        1951     1894.27     2079.76
         298     1842.98        1973     1901.86     2103.02
         299     1848.64        1979     1909.45     2109.36
         300     1856.18        1987     1917.04     2117.82
         301     1861.84        1993     1924.64     2124.16
         302     1865.61        1997     1932.24     2128.39
         303     1867.49        1999     1939.84     2130.51
         304     1871.27        2003     1947.44     2134.73
         305     1878.81        2011     1955.05     2143.19
         306     1884.47        2017     1962.66     2149.53
         307      1893.9        2027     1970.27      2160.1
         308     1895.78        2029     1977.89     2162.22
         309     1905.21        2039     1985.51     2172.79
         310     1918.41        2053     1993.13     2187.59
         311     1927.84        2063     2000.77     2198.16
         312      1933.5        2069      2008.4      2204.5
         313     1944.82        2081     2016.04     2217.18
         314     1946.71        2083     2023.68     2219.29
         315     1950.48        2087     2031.33     2223.52
         316     1952.36        2089     2038.97     2225.64
         317      1961.8        2099     2046.62      2236.2
         318     1973.11        2111     2054.28     2248.89
         319        1975        2113     2061.93        2251
         320     1990.09        2129     2069.59     2267.91
         321     1991.98        2131     2077.26     2270.02
         322     1997.64        2137     2084.93     2276.36
         323     2001.41        2141      2092.6     2280.59
         324      2003.3        2143     2100.26      2282.7
         325     2012.73        2153     2107.94     2293.27
         326     2020.28        2161     2115.62     2301.72
         327     2037.26        2179      2123.3     2320.74
         328      2059.9        2203        2131      2346.1
         329     2063.68        2207      2138.7     2350.32
         330     2069.34        2213      2146.4     2356.66
         331     2076.89        2221     2154.11     2365.11
         332     2091.98        2237     2161.82     2382.02
         333     2093.87        2239     2169.54     2384.13
         334     2097.64        2243     2177.25     2388.36
         335     2105.19        2251     2184.97     2396.81
         336     2120.29        2267      2192.7     2413.71
         337     2122.18        2269     2200.42     2415.82
         338     2125.95        2273     2208.15     2420.05
         339      2133.5        2281     2215.89      2428.5
         340     2139.17        2287     2223.62     2434.83
         341     2144.83        2293     2231.36     2441.17
         342      2148.6        2297      2239.1      2445.4
         343     2159.93        2309     2246.84     2458.07
         344     2161.82        2311     2254.59     2460.18
         345     2182.58        2333     2262.34     2483.42
         346     2188.24        2339      2270.1     2489.76
         347     2190.13        2341     2277.86     2491.87
         348     2195.79        2347     2285.62     2498.21
         349     2199.57        2351     2293.38     2502.43
         350     2205.23        2357     2301.15     2508.77
         351     2218.45        2371     2308.92     2523.55
         352     2224.11        2377     2316.69     2529.89
         353     2227.89        2381     2324.47     2534.11
         354     2229.77        2383     2332.24     2536.23
         355     2235.44        2389     2340.02     2542.56
         356     2239.21        2393      2347.8     2546.79
         357     2244.88        2399     2355.58     2553.12
         358     2256.21        2411     2363.37     2565.79
         359     2261.87        2417     2371.16     2572.13
         360     2267.54        2423     2378.96     2578.46
         361     2280.75        2437     2386.75     2593.25
         362     2284.53        2441     2394.55     2597.47
         363     2290.19        2447     2402.36     2603.81
         364     2301.52        2459     2410.16     2616.48
         365     2309.08        2467     2417.97     2624.92
         366     2314.74        2473     2425.79     2631.26
         367     2318.52        2477      2433.6     2635.48
         368     2343.07        2503     2441.43     2662.93
         369     2360.07        2521     2449.26     2681.93
         370     2369.51        2531      2457.1     2692.49
         371     2377.06        2539     2464.94     2700.94
         372     2380.84        2543     2472.78     2705.16
         373     2386.51        2549     2480.62     2711.49
         374      2388.4        2551     2488.47      2713.6
         375     2394.06        2557     2496.31     2719.94
         376     2414.84        2579     2504.17     2743.16
         377     2426.17        2591     2512.03     2755.83
         378     2428.06        2593     2519.89     2757.94
         379     2443.17        2609     2527.75     2774.83
         380     2450.73        2617     2535.62     2783.27
         381     2454.51        2621      2543.5     2787.49
         382     2465.84        2633     2551.37     2800.16
         383     2479.07        2647     2559.25     2814.93
         384     2488.51        2657     2567.14     2825.49
         385      2490.4        2659     2575.02      2827.6
         386     2494.18        2663     2582.91     2831.82
         387     2501.74        2671      2590.8     2840.26
         388     2507.41        2677     2598.69     2846.59
         389     2513.08        2683     2606.59     2852.92
         390     2516.85        2687     2614.48     2857.15
         391     2518.74        2689     2622.38     2859.26
         392     2522.52        2693     2630.28     2863.48
         393     2528.19        2699     2638.18     2869.81
         394     2535.75        2707     2646.08     2878.25
         395     2539.53        2711     2653.99     2882.47
         396     2541.42        2713     2661.89     2884.58
         397     2547.09        2719      2669.8     2890.91
         398     2556.53        2729     2677.71     2901.47
         399     2558.42        2731     2685.63     2903.58
         400     2567.87        2741     2693.54     2914.13
         401     2575.43        2749     2701.46     2922.57
         402     2579.21        2753     2709.38     2926.79
         403     2592.44        2767     2717.31     2941.56
         404     2601.89        2777     2725.24     2952.11
         405     2613.23        2789     2733.17     2964.77
         406     2615.11        2791      2741.1     2966.89
         407     2620.78        2797     2749.04     2973.22
         408     2624.56        2801     2756.98     2977.44
         409     2626.45        2803     2764.92     2979.55
         410     2641.57        2819     2772.86     2996.43
         411      2654.8        2833     2780.81      3011.2
         412     2658.58        2837     2788.76     3015.42
         413     2664.25        2843     2796.71     3021.75
         414     2671.81        2851     2804.67     3030.19
         415     2677.48        2857     2812.63     3036.52
         416     2681.26        2861     2820.58     3040.74
         417     2698.28        2879     2828.55     3059.72
         418     2705.84        2887     2836.52     3068.16
         419     2715.29        2897     2844.49     3078.71
         420     2720.96        2903     2852.46     3085.04
         421     2726.63        2909     2860.44     3091.37
         422     2734.19        2917     2868.42     3099.81
         423     2743.64        2927      2876.4     3110.36
         424     2754.99        2939     2884.38     3123.01
         425     2768.22        2953     2892.37     3137.78
         426        2772        2957     2900.37        3142
         427     2777.67        2963     2908.36     3148.33
         428     2783.34        2969     2916.36     3154.66
         429     2785.23        2971     2924.35     3156.77
         430      2811.7        2999     2932.36      3186.3
   prime fac     R-S low           p     log(p#)     R-S high

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